Shearing wave

A shearing wave is a local disturbance whose wavevector evolves under background shear. In a Keplerian sheet, $k_x(t)=k_x(0)+(3/2)\Omega k_{y}t$.

For background shear ${\mathbf{u}}_0=-2Ax{\mathbf{e}}_y$, take ${\mathbf{u}}^{\prime }=\mathrm{Re}[\mathbf{v}(t)e^{i\mathbf{k}(t)\cdot \mathbf{r}}]$ with real $\mathbf{k}$ and $\mathbf{k}\cdot \mathbf{v}=0$. Then ${\mathbf{u}}^{\prime }\cdot \nabla {\mathbf{u}}^{\prime }=0$ pointwise, including conjugate terms. Choosing ${\dot{k}}_x=2A{k}_y$, ${\dot{k}}_y={\dot{k}}_z=0$ cancels background phase advection. The resulting linear amplitude equations are therefore exact finite-amplitude solutions of the incompressible Navier-Stokes equation in this unbounded or compatible shearing-periodic setting.

// Target: gravitational-instability-of-an-astrophysical-disk.bigb

In an incompressible shearing sheet, a single real-wavevector velocity and magnetic Fourier mode transverse to that wavevector has identically zero quadratic self-advection and cross-advection. Advecting its phase by background shear and stretching its spatially uniform field give the displayed evolution laws. Their opposite contributions keep the Alfvén frequency constant. The resulting viscous and resistive wave-amplitude equations are exact finite-amplitude solutions; physical pressure adjusts to the quadratic magnetic-pressure terms.

In an incompressible shearing sheet with background velocity $-2Ax{\mathbf{e}}_y$, take both the perturbation velocity and the entire magnetic field to be real single Fourier modes with a common real wavevector and transverse amplitudes. Then every quadratic directional derivative vanishes, including terms involving complex conjugates. There is no uniform background magnetic field and consequently no linear magnetic-tension coupling between the amplitudes. Advecting the phase requires ${\dot{k}}_x=2A{k}_y$, ${\dot{k}}_y={\dot{k}}_z=0$. The velocity amplitude obeys $\dot{\tilde{\mathbf{v}}}-2A{\tilde{v}}_x{\mathbf{e}}_y+2\mathbf{\Omega }\times \tilde{\mathbf{v}}=-i\mathbf{k}\tilde{\psi }-\nu k^2\tilde{\mathbf{v}}$; the magnetic amplitude obeys the displayed equation. The modified pressure includes magnetic pressure; the physical pressure can contain a mean and second harmonic even when the modified pressure is a single mode.

For a zero-mean magnetic shearing wave, the real wavevector satisfies $k_x=k_{x0}+2A{k}_{y}t$, with $k_y,k_z$ constant. Dotting the velocity-amplitude equation with its complex conjugate eliminates Coriolis force and pressure work, giving $d|\tilde{\mathbf{v}}{|}^2/dt=4A\mathrm{Re}({\tilde{v}}_x{\tilde{v}}_y^{\ast })-2\nu k^2|\tilde{\mathbf{v}}{|}^2$. The inequality $2|v_x{v}_y|\le |\mathbf{v}{|}^2$ proves the displayed bound; the magnetic bound has the same form with $\eta$ replacing $\nu$. NowFor $A{k}_y\ne 0$ and positive kinematic viscosity and magnetic diffusivity, the negative cubic exponent dominates the linear shear-work bound, so both amplitudes tend to zero. Transient growth is allowed. Positivity matters: an inviscid vertical velocity mode with $k_z=0$ can remain constant, and an ideal vertical magnetic mode with $k_z=0$ can also remain constant.

For $k_z=v_z=0$ and $k_y\ne 0$, define $Z=i(k_x{v}_y-k_y{v}_x)$. Uniform rotation does not enter its evolution: $\dot{Z}=-\nu k^{2}Z$, $v_x=i{k}_{y}Z/k^2$ and $v_y=-i{k}_{x}Z/k^2$. With $T=k_x/k_y$ and $\mathrm{Re}=A/(\nu k_y^2)$, the perturbation energy ratio is $E(T)/E(T_0)=[(1+T_0^2)/(1+T^2)]\exp [-(T+T^3/3-T_0-T_0^3/3)/\mathrm{Re}]$. Leading waves can gain energy as shear reduces their wavevector magnitude, before viscosity and subsequent winding damp them.

// Target: hydrodynamic-stability.bigb

For the shearing-wave energy profile, stationary points satisfy $-2\mathrm{Re}T=(1+T^2{)}^2$. At large Reynolds number, the initial minimum is at $T_{-}\sim -(2\mathrm{Re}{)}^{1/3}$ and the subsequent maximum is at $T_{+}\sim -1/(2\mathrm{Re})$. Their energy ratio is $G_{\max }\sim (2\mathrm{Re}{)}^{2/3}{e}^{-2/3}$. The Reynolds number here uses $A$, half the background shear rate; using $2A$ instead changes the coefficient convention.

// Target: astrophysics.bigb

For a solenoidal velocity amplitude $\mathbf{v}$ with $\mathbf{k}=(k_x,-\Omega t{k}_x,k_z)$ and $\dot{\mathbf{v}}+\Omega v_y\hat{\mathbf{x}}=-i\mathbf{k}q$, differentiating $\mathbf{k}\cdot \mathbf{v}=0$ gives $q=2i\Omega k_x{v}_y/K^2$, where $K^2=|\mathbf{k}{|}^2$. Thus ${\dot{v}}_y=2\Omega k_x{k}_y{v}_y/K^2$ and ${\dot{v}}_z=2\Omega k_x{k}_z{v}_y/K^2$. For the real helicity quantity $H=i{k}_x(v_y{v}_z^{\ast }-v_z{v}_y^{\ast })$, the mixed forcing terms cancel, leaving $\dot{H}=2\Omega k_x{k}_{y}H/K^2=-{\dot{K}}^{2}H/K^2$. ConsequentlyThe invariant is a consequence of inviscid linear shear dynamics, not a claim that $H$ itself is constant under the shear.

A shearing-wave ansatz in a uniform inviscid isothermal shearing sheet removes background advection when the radial wavevector grows as $k_x(t)=k_x(0)+S{k}_{y}t$. The razor-thin disk Poisson kernel reduces the pressure-plus-gravity amplitude to $[c_s^2-2\pi G{\Sigma }_0/k]{\tilde{\Sigma }}^{\prime }/{\Sigma }_0$. Continuity couples it to $i{k}_x{v}_x+i{k}_y{v}_y$, while the two momentum equations have rotational couplings $-2\Omega v_y$ and $(2\Omega -S)v_x$. The linearized vortensity amplitude remains constant, because the barotropic pressure and gravitational forces have zero curl and the background vortensity is uniform.

The shearing-wave ansatz writes a perturbation as a time-dependent amplitude times $\exp [i{k}_x(t)x+i{k}_{y}y]$, choosing $k_x(t)$ so that background advection does not leave an explicit factor of position in the amplitude equations.

For an unforced inertial-acoustic disturbance, the shearing-wave amplitude obeys a time-dependent oscillator equation whose frequency depends on $k_x(t)=k_x(0)+S{k}_{y}t$. Fourier transformation of the corresponding stationary forced equation turns radial wavenumber into the oscillator's time-like coordinate.

During the swing of a shearing wave, the radial wavenumber passes through zero and the disturbance changes from leading to trailing before winding ever more tightly.